In this talk, we introduce expander graphs and illustrate some connections with number theory. We start from first principles in graph theory, Cayley graphs and the basic theory of expanders. Next, we describe the spectral properties of the discrete Laplacian from a geometric perspective and outline a basic construction of families of expander graphs by Lubotzky, Phillips and Sarnak. Time permitting, we include at the end an overview of the recent results on expansion in linear groups over finite fields and some of their applications to finiteness statements in arithmetic geometry. The major part of the talk is mostly self-contained, assuming only basic linear algebra.